HAL Id: hal-01492048
https://hal.archives-ouvertes.fr/hal-01492048
Submitted on 17 Mar 2017
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Close to Global Equilibrium
Marc Briant, Esther Daus
To cite this version:
Marc Briant, Esther Daus. The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium. Archive for Rational Mechanics and Analysis, Springer Verlag, 2016, 222 (3), pp.1367-1443. �10.1007/s00205-016-1023-x�. �hal-01492048�
MARC BRIANT AND ESTHER S. DAUS
Abstract. We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the 3-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in L1
vL∞x(m), where m ∼ (1 + |v| k
) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an L2
− L∞ theory `a la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (e.g. Carleman representa-tion, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the L1vL∞x framework is dealt with for any k > k0, recov-ering the optimal physical threshold of finite energy k0= 2 in the particular case of a multi-species hard spheres mixture with same masses.
Keywords: Multi-species mixture; Boltzmann equation; Spectral gap; Perturba-tive theory; Convergence to equilibrium; L2− L∞ theory, Carleman representation,
Povzner inequality.
Acknowledgements: The second author wants to thank Ansgar J¨ungel for his valuable help.
Contents
1. Introduction 2
2. Main results 10
3. Spectral gap for the linear operator in L2 v µ
−1/2
12
4. L2 theory for the linear part with maxwellian weight 21
5. L∞ theory for the linear part with maxwellian weight 36
6. The full nonlinear equation in a perturbative regime 50
7. Ethical Statement 69
References 70
The first author was partly supported by the 150thAnniversary Postdoctoral Mobility Grant of the London Mathematical Society and the Division of Applied Mathematics at Brown University. The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P24304, P27352, and W1245, and the Austrian-French Program of the Austrian Exchange Service ( ¨OAD).
1. Introduction
The present work establishes existence, uniqueness, positivity and exponential trend to equilibrium for the multi-species Boltzmann equation close to equilibrium, which is used in physics and biology to model the evolution of a dilute gaseous mixture with different masses. The physically most relevant space for such a Cauchy theory is the space of density functions that only have finite mass and energy, which are the first and second moments in the velocity variable. This present article proves the result in the space L1
vL ∞ x (1+|v|
k
) for any k > k0, where k0is an explicit threshold
depending heavily on the differences of the masses, recovering the physically optimal threshold k0 = 2 when all the masses of the mixture are the same and the particles
are approximated to be hard spheres.
We are thus interested in the evolution of a dilute gas on the torus T3composed of
N different species of chemically non-reacting mono-atomic particles, which can be modeled by the following system of Boltzmann equations, stated on R+× T3× R3,
(1.1) ∀ 1 6 i 6 N, ∂tFi(t, x, v) + v· ∇xFi(t, x, v) = Qi(F)(t, x, v)
with initial data
∀ 1 6 i 6 N, ∀(x, v) ∈ T3× R3, F
i(0, x, v) = F0,i(x, v).
Note that the distribution function of the system is given by the vector F = (F1, . . . , FN), with Fi describing the ith species at time t, position x and velocity
v.
The Boltzmann operator Q(F) = (Q1(F), . . . , QN(F)) is given for all i by
Qi(F) = N
X
j=1
Qij(Fi, Fj),
where Qij describes interactions between particles of either the same (i = j) or of
different (i6= j) species and are local in time and space. Qij(Fi, Fj)(v) = Z R3×S2 Bij(|v − v∗|, cos θ) h Fi0Fj0∗− FiFj∗ i dv∗dσ,
where we used the shorthands F0
i = Fi(v0), Fi = Fi(v), F 0∗ j = Fj(v∗0) and Fj∗ = Fj(v∗). v0 = 1 mi+ mj (miv + mjv∗+ mj|v − v∗|σ) v0∗ = 1 mi+ mj (miv + mjv∗− mi|v − v∗|σ) , and cos θ = v− v∗ |v − v∗| , σ .
Note that these expressions imply that we deal with gases where only binary elastic collisions occur (the mass mi of all molecules of species i remains the same, since
there is no reaction). Indeed, v0 and v0
∗ are the velocities of two molecules of species
i and j before collision giving post-collisional velocities v and v∗ respectively, with
conservation of momentum and kinetic energy:
miv + mjv∗ = miv0+ mjv∗0, 1 2mi|v| 2 + 1 2mj|v∗| 2 = 1 2mi|v 0 |2+1 2mj|v 0 ∗| 2 . (1.2)
The collision kernels Bij are nonnegative, moreover they contain all the
informa-tion about the interacinforma-tion between two particles and are determined by physics. We mention at this point that one can derive this type of equations from Newtonian mechanics at least formally in the case of single species [11][12]. The rigorous va-lidity of the mono-species Boltzmann equation from Newtonian laws is known for short times (Landford’s theorem [28] or more recently [17][33]).
1.1. The perturbative regime and its motivation. Using the standard changes of variables (v, v∗) 7→ (v0, v0∗) and (v, v∗) 7→ (v∗, v) (note the lack of symmetry
be-tween v0 and v0
∗ compared to v for the second transformation due to different masses)
together with the symmetries of the collision operators (see [11][12][37] among oth-ers and [14][13] and in particular [7] for multi-species specifically), we recover the following weak forms:
Z R3 Qij(Fi, Fj)(v)ψi(v) dv = Z R6 Z S2 Bij(|v − v∗|, cos(θ))FiFj∗(ψ 0 i− ψi) dσdvdv∗ and Z R3 Qij(Fi, Fj)(v)ψi(v) dv + Z R3 Qji(Fj, Fi)(v)ψj(v) dv = −1 2 Z R6 Z S2 Bij(|v − v∗|, cos(θ)) Fi0Fj∗− FiFj∗ ψ0i+ ψ0∗j − ψi− ψ∗j dσdvdv∗. (1.3) Thus (1.4) N X i,j=1 Z R3 Qij(Fi, Fj)(v)ψi(v) dv = 0
if and only if ψ(v) belongs to Spane1, . . . , eN, v1m, v2m, v3m,|v| 2
m , where ek
stands for the kth unit vector in RN and m = (m
1, . . . , mN). The fact that we need
to sum over i has interesting consequences and implies a fundamental difference compared with the single-species Boltzmann equation. In particular it implies con-servation of the total number density c∞,i of each species, of the total momentum of
the gas ρ∞u∞ and its total energy 3ρ∞θ∞/2:
∀t > 0, c∞,i = Z T3×R3 Fi(t, x, v) dxdv (1 6 i 6 N) u∞= 1 ρ∞ N X i=1 Z T3×R3 mivFi(t, x, v) dxdv θ∞= 1 3ρ∞ N X i=1 Z T3×R3 mi|v − u∞|2Fi(t, x, v) dxdv, (1.5)
where ρ∞ = PNi=1mic∞,i is the global density of the gas. Note that this already
shows intricate interactions between each species and the total mixture itself. The operator Q = (Q1, . . . , QN) also satisfies a multi-species version of the
F= (F1, . . . , FN) being the maximum of the Boltzmann entropy, has the form of a
local Maxwellian, that is
∀ 1 6 i 6 N, Fi(t, x, v) = cloc,i(t, x) mi 2πkBθloc(t, x) 3/2 exp " −mi|v − u loc(t, x)|2 2kBθloc(t, x) # . Here kB is the Boltzmann constant and, denoting the total local mass density by
ρloc =PNi=1micloc,i, we used the following local definitions
∀ 1 6 i 6 N, cloc,i(t, x) = Z R3 Fi(t, x, v) dv, uloc(t, x) = 1 ρloc N X i=1 Z R3 mivFidv, θloc(t, x) = 1 3ρloc N X i=1 Z R3 mi|v − uloc|2Fidv.
On the torus, this multi-species H-theorem also implies that the global equilib-rium, i.e. a stationary solution F to (1.1), associated to the initial data F0(x, v) =
(F0,1, . . . , F0,N) is uniquely given by the global Maxwellian
∀ 1 6 i 6 N, Fi(t, x, v) = Fi(v) = c∞,i mi 2πkBθ∞ 3/2 exp " −mi|v − u ∞|2 2kBθ∞ # . By translating and rescaling the coordinate system we can always assume that u∞=
0 and kBθ∞ = 1 so that the only global equilibrium is the normalized Maxwellian
(1.6) µ= (µi)16i6N with µi(v) = c∞,i
mi 2π
3/2
e−mi|v|22 .
The aim of the present article is to construct a Cauchy theory for the multi-species Boltzmann equation (1.1) around the global equilibrium µ. In other terms we study the existence, uniqueness and exponential decay of solutions of the form Fi(t, x, v) = µi(v) + fi(t, x, v) for all i.
Under this perturbative regime, the Cauchy problem amounts to solving the per-turbed multi-species Boltzmann system of equations
(1.7) ∂tf + v· ∇xf = L(f ) + Q(f ),
or equivalently in the non-vectorial form
∀ 1 6 i 6 N, ∂tfi+ v· ∇xfi = Li(f ) + Qi(f ),
where f = (f1, . . . , fN) and the operator L = (L1, . . . , LN) is the linear Boltzmann
operator given for all 1 6 i 6 N by Li(f ) = N X j=1 Lij(fi, fj), with Lij(fi, fj) = Qij(µi, fj) + Qij(fi, µj).
Since we are looking for solutions F preserving individual mass, total momentum and total energy (1.5) we have the equivalent perturbed conservation laws for f =
F− µ which are given by ∀t > 0, 0 = Z T3×R3 fi(t, x, v) dxdv (1 6 i 6 N) 0 = N X i=1 Z T3×R3 mivfi(t, x, v) dxdv 0 = N X i=1 Z T3×R3 mi|v| 2 fi(t, x, v) dxdv. (1.8)
1.2. Notations and assumptions on the collision kernel. First, to avoid any confusion, vectors and vector-valued operators in RN will be denoted by a bold
symbol, whereas their components by the same indexed symbol. For instance, W represents the vector or vector-valued operator (W1, . . . , WN).
We define the Euclidian scalar product in RN weighted by a vector W by
hf, giW = N
X
i=1
figiWi.
In the case W = 1 = (1, . . . , 1) we may omit the index 1.
Function spaces. We define the following shorthand notation hvi =
q
1 +|v|2.
The convention we choose is to index the space by the name of the concerned variable, so we have for p in [1, +∞]
Lp[0,T ] = Lp([0, T ]) , Lpt = L p R+ , Lpx = L p T3 , Lpv = L p R3 . For W = (W1, . . . , WN) : R3 −→ R+ a strictly positive measurable function in v,
we will use the following vector-valued weighted Lebesgue spaces defined by their norms kfkL2 v(W) = N P i=1kf ik2L2 v(Wi) 1/2 , kfikL2 v(Wi) =kfiWi(v)kL2v, kfkL2 x,v(W) = N P i=1kf ik2L2 x,v(Wi) 1/2 , kfikL2 x,v(Wi)= kfikL2 xWi(v) L2 v, kfkL∞ x,v(W) = N P i=1kf ikL∞
x,v(Wi), kfikL∞x,v(Wi)= sup
(x,v)∈T3×R3 |fi (x, v)| Wi(v), kfkL1 vL∞x (W) = N P i=1kf ikL1 vL∞x(Wi), kfikL1vL∞x (Wi) = sup x∈T3|f i(x, v)| Wi(v) L1 v . Note that L2
v(W) and L2x,v(W) are Hilbert spaces with respect to the scalar products
hf, giL2 v(W) = N X i=1 hfi, giiL2 v(Wi) = N X i=1 Z R3 figiWi2dv, hf, giL2 x,v(W) = N X i=1 hfi, giiL2 x,v(Wi) = N X i=1 Z T3×R3 figiWi2dxdv.
Assumptions on the collision kernel.
We will use the following assumptions on the collision kernels Bij.
(H1) The following symmetry holds
Bij(|v − v∗|, cos θ) = Bji(|v − v∗|, cos θ) for 1 ≤ i, j ≤ N.
(H2) The collision kernels decompose into the product
Bij(|v − v∗|, cos θ) = Φij(|v − v∗|)bij(cos θ), 1≤ i, j ≤ N,
where the functions Φij ≥ 0 are called kinetic part and bij ≥ 0 angular part.
This is a common assumption as it is technically more convenient and also covers a wide range of physical applications.
(H3) The kinetic part has the form of hard or Maxwellian (γ = 0) potentials, i.e. Φij(|v − v∗|) = CijΦ|v − v∗|γ, CijΦ > 0, γ ∈ [0, 1], ∀ 1 6 i, j 6 N.
(H4) For the angular part, we assume a strong form of Grad’s angular cutoff (first introduced in [19]), that is: there exist constants Cb1, Cb2 > 0 such that for
all 1≤ i, j ≤ N and θ ∈ [0, π],
0 < bij(cos θ)≤ Cb1| sin θ| | cos θ|, b0ij(cos θ)≤ Cb2.
Furthermore, Cb := min 1≤i≤Nσ1,σ2∈Sinf 2 Z S2 minbii(σ1· σ3), bii(σ2· σ3) dσ3 > 0.
We emphasize here that the important cases of Maxwellian molecules (γ = 0 and b = 1) and of hard spheres (γ = b = 1) are included in our study. We shall use the standard shorthand notations
(1.9) b∞ij =kbijkL∞
[−1,1] and lbij =kb ◦ coskL 1 S2
.
1.3. Novelty of this article. As mentioned previously, the present work proves the existence, uniqueness, positivity and exponential trend to equilibrium for the full nonlinear multi-species Boltzmann equation (1.1) in L1
vL ∞
x hvik with the explicit
threshold k > k0 defined in Lemma 6.3, when the initial data F0 is close enough to
the global equilibrium µ. This is equivalent to solving the perturbed equation (1.7) for small f0. This perturbative Cauchy theory for gaseous mixtures is completely
new.
Moreover, one of the major contributions of the present article is to combine and adapt several very recent strategies, combined with new hypocoercivity estimates, in order to develop a new constructive approach that allows to deal with polynomial weights without requiring any spatial Sobolev regularity. This is new even in the mono-species case even though the final result we obtain has recently been proved for the mono-species hard sphere model [22]) (which we therefore also extend to more general hard and Maxwellian potential kernels.).
Also, as a by-product, we prove explicitly that the linear operator L− v · ∇x
gen-erates a strongly continuous semigroup with exponential decay both in L2 x,v µ
−1/2 and in L∞
x,v hviβµ
−1/2; such constructive and direct results on the torus are new to
At last, we derive new estimates in order to deal with different masses and the multi-species cross-interaction operators, and we also extend recent mono-species estimates to more general collision kernels. Note that the asymmetry of the elastic collisions requires to derive a new description of Carleman’s representation of the Boltzmann operator as well as new Povzner-type inequalities suitable for this lack of symmetry.
1.4. State of the art and strategy. Very little is known about any rigorous Cauchy theory for multi-species gases with different masses. We want to mention [6], where a compactness result for the linear operator K := L + ν was proved in L2
v(µ−1/2). For multi-species gases with same masses, the recent work [13] proved
that the operator L has a spectral gap in L2 v µ
−1/2 and obtained an a priori
expo-nential convergence to equilibrium for the perturbed equation (1.7) in H1 x,v µ
−1/2.
We emphasize here that [13] only studied the case of same masses mi = mj for all i,
j. On the contrary, the single-species Boltzmann equation in the perturbative regime around a global Maxwellian has been extensively studied over the past fifty years (see [35] for an exhaustive review). Starting with Grad [21], the Cauchy problem has been tackled in L2 vHxs µ −1/2 spaces [34], in Hs x,v µ −1/2(1 + |v|)k [24][38] was then extended to Hs x,v µ
−1/2 where an exponential trend to equilibrium has also been
obtained [31][25]. Recently, [22] proved existence and uniqueness for single-species Boltzmann equation in more the general spaces (Wα,1
v ∩ Wvα,q) Wxβ,p (1 +|v|)k for
α 6 β and β and k large enough with explicit thresholds. The latter paper thus includes L1
vL∞x hvik. All the results presented above hold in the case of the torus
for hard and Maxwellian potentials. We refer the reader interested in the Cauchy problem to the review [35].
All the works mentioned above involve to working in spaces with derivatives in the space variable x (we shall discuss some of the reasons later) with exponential weight. The recent breakthrough [22] gets rid of both the Sobolev regularity and the exponential weight but uses a new extension method which still requires to have a well-established linear theory in Hs
x,v µ −1/2.
Our strategy can be decomposed into four main steps and we now describe each of them and their link to existing works.
Step 1: Spectral gap for the linear operator in L2 v µ
−1/2. It has been
known for long that the single-species linear Boltzmann operator L is a self-adjoint non positive linear operator in the space L2
v µ−1/2. Moreover it has a spectral gap
λ0. This has been proved in [10][19][20] with non constructive methods for hard
potential with cutoff and in [4][5] in the Maxwellian case. These results were made constructive in [1][30] for more general collision operators. One can easily extend this spectral gap to Sobolev spaces Hs
v µ
−1/2 (see for instance [22] Section 4.1).
Recently, [13] proved the existence of an explicit spectral gap for the operator L for multi-species mixtures where all the masses are the same (mi = mj). Our
constructive spectral gap estimate in L2 v µ
−1/2 closely follows their methods that
consist in proving that the cross-interactions between different species do not perturb too much the spectral gap that is known to exist for the diagonal operator Lii
(single-species operators). We emphasize here that not only we adapt the methods of [13] to fit the different masses framework but we also derive estimates on the collision
frequencies that allow us to get rid of their strong requirement on the collision kernels: Bij 6 βBiifor all i, j. The latter assumption is indeed physically irrelevant
in our framework. Step 2: L2
x,v µ −1/2
theory for the full perturbed linear operator. The next step is to prove that the existence of a spectral gap for L in the sole veloc-ity variable can be transposed to L2
x,v µ
−1/2 when one adds the skew-symmetric
transport operator −v · ∇x. In other words, we prove that G = L− v · ∇x generates
a strongly continuous semigroup in L2
x,v µ−1/2 with exponential decay.
One thus wants to derive an exponential decay for solutions to the linear perturbed Boltzmann equation
∂tf + v· ∇xf = L (f ) .
A straightforward use of the spectral gap λL of L shows for such a solution
d dt kfk 2 L2 x,v(µ−1/2) 6 −2λLkf − πL(f )k 2 L2 x,v(µ−1/2) ,
where πL stands for the orthogonal projection in L2v µ
−1/2 onto the kernel of the
operator L. This inequality exhibits the hypocoercivity of L. Roughly speaking, the exponential decay in L2
x,v µ
−1/2 would follow for solutions f if the microscopic part
π⊥
L(f ) = f − πL(f ) controls the fluid part which has the following form (see Section
3) ∀1 6 i 6 N, πL(f )i(t, x, v) = ai(t, x) + b(t, x)· v + c(t, x)|v| 2− 3m−1 i 2 miµi(v),
where ai(t, x), c(t, x)∈ R and b(t, x) ∈ R3are the coordinates of an orthogonal basis.
The standard strategies in the case of the single-species Boltzmann equation are based on higher Sobolev regularity either from hypocoercivity methods [31] or elliptic regularity of the coefficients a, b and c [23][25]. Roughly speaking one has [23][25] (1.10) ∆πL(f )∼ ∂2πL⊥f + higher order terms,
which can be combined with elliptic estimates to control the fluid part by the mi-croscopic part in Sobolev spaces Hs. Our main contribution to avoid involving
high regularity is based on an adaptation of the recent work [15] (dealing with the single-species Boltzmann equation with diffusive boundary conditions). The key idea consists in integrating against test functions that contains a weak version of the elliptic regularity of a(t, x), b(t, x) and c(t, x). Basically, the elliptic regularity of πL(f ) will be recovered thanks to the transport part applied to these test functions
while on the other side L encodes the control by π⊥ L (f ).
It has to be emphasized that thanks to boundary conditions, [15] only needed the conservation of mass whereas in our case this “weak version” of estimates (1.10) strongly relies on all the conservation laws. The choice of test functions thus has to take into account the delicate interaction between each species and the total mixture we already pointed out. This leads to intricate technicalities since for each species we need to deal with different reference rates of decay mi. Finally, our proof also
involves elliptic regularity in negative Sobolev spaces to deal with ∂ta, ∂tb and ∂tc.
Step 3: L∞
x,v hviβµ−1/2
theory for the full nonlinear equation. Thanks to the first two steps we have a satisfactory L2 semigroup theory for the full linear
equation (see [11][12] or [37] for instance), the underlying L2
x,v-norm is not an
alge-braic norm for the nonlinear operator Q whereas the L∞
x,v-norm is.
The key idea of proving a semigroup property in L∞ is thanks to an L2 − L∞
theory “`a la Guo” [26], where the L∞-norm will be controlled by the L2-norm along
the characteristics. As we shall see, each component Li can be decomposed into
Li = Ki − νi where νi(f ) = νi(v)fi is a multiplicative operator. If we denote by
SG(t) the semigroup generated by G = L− v · ∇x, we have the following implicit
Duhamel representation of its ith component along the characteristics
SG(t)i = e−νi(v)t+
Z t
0
e−νi(v)(t−s)K
i[SG(s)] ds.
Following the idea of Vidav [36] and later used in [26], an iteration of the above should yield a certain compactness property. Hiding here all the cross-interactions, we end up with SG(t) =e−ν(v)t+ Z t 0 e−ν(v)(t−s)Ke−ν(v)sds + Z t 0 Z s 0 e−ν(v)(t−s)Ke−ν(v)(s−s1)K[S G(s1)] ds1ds.
We shall prove that K is compact and is a kernel operator. The first two terms will be easily estimated and the last term will be roughly of the form
Z t 0 Z s 0 Z v1,v2bounded|S G(s1, x− (t − s)v − (s − s1)v1, v2| dv2dv1ds1ds.
The double integration implies that v1 and v2 are independent and we can thus
perform a change of variables which changes the integral in v1 into an integral over
T3 that we can bound thanks to the previous L2 theory. For integrability reasons, this third step actually proves that G generates a strongly continuous semigroup with exponential decay in L∞
hviβµ−1/2 for β > 3/2.
Our work provides two key contributions to prove the latter result. First, to prove the desired pointwise estimate for the kernel of K, we need to give a new representation of the operator in terms of the parameters (v0, v0
∗) instead of (v∗, σ).
In the single-species case, such a representation is the well-known Carleman rep-resentation [10] and requires integration onto the so-called Carleman hyperplanes hv0
− v, v0
∗ − vi = 0. However, when particles have different masses, the lack of
symmetry between v0 and v0
∗ compared to v obliges us to derive new Carleman
ad-missible sets (some become spheres). Second, the decay of the exponential weight differs from one species to the other. To obtain estimates that are similar to the case of single-species we exhibit the property that K mixes the exponential rate of decay among the cross-interaction between species. This enables us to close the L∞
estimate for the first two terms of the iterated Duhamel representation. Step 4: Extension to polynomial weights and L1
vL ∞
x space. To conclude
the present study, we develop an analytic and nonlinear version of the recent work [22], also recently adapted in a nonlinear setting [8]. The main strategy is to find a decomposition of the full linear operator G into G1 + A. We shall prove that
G1 acts like a small perturbation of the operator Gν =−v · ∇x− ν(v) and is thus
the operator A allows us to decompose the perturbative equation (1.7) into a system of differential equations
∂tf1 = G1(f1) + Q(f1+ f2, f1+ f2)
∂tf2+ v· ∇xf2 = L (f2) + A (f1)
The first equation is solved in L∞
x,v(m) or L1vL∞x (m) with the initial data f0 thanks
to the hypodissipativity of G1. The regularity of A (f1) allows us to use Step 3 and
thus solve the second equation with null initial data in L∞
x,v hviβµ−1/2). First, the
existence of a solution to the system having exponential decay is obtained thanks to an iterative scheme combined with new estimates on the multi-species operators G1 and A. Then uniqueness follows a new stability estimate in an equivalent norm
(proposed in [22]), that fits the dissipativity of the semigroup generated by G. Finally, positivity of the unique solution comes from a different iterative scheme.
In the case of the single-species Boltzmann equation, the less regular weight m(v) one can achieve with this method is determined by the hypodissipative property of G1 and gives m =hvik with k > 2, which is indeed obtained also in the multi-species
framework of same masses. In the general case of different masses, the threshold k0
is more intricate (see Theorem2.2), since it also depends on the different masses mi.
1.5. Organisation of the paper. The paper follows exactly the four steps de-scribed above.
Section 2 gives a precise statement of the main theorems that will be proved in this work and the rest of the article is dedicated to the proof of these theorems.
Section3deals with the spectral gap of L. The semigroup property in L2
x,v µ−1/2
is treated in Section4. This property is then passed on to L∞
x,v hviβµ
−1/2 in Section
5.
At last, we work out the Cauchy problem for the full nonlinear equation in Section
6.
2. Main results
As explained in the introduction, the ultimate goal of this article is a full per-turbative Cauchy theory for the multi-species Boltzmann equation (1.1). Along the way, we shall also prove the following important results about the linear perturbed operator L− v · ∇x.
Theorem 2.1. Let the collision kernels Bij satisfy assumptions (H1)− (H4). Then
the following holds.
(i) The operator L is a closed self-adjoint operator in L2 v µ
−1/2 and there exists
λL > 0 such that ∀f ∈ L2 v µ −1/2 , hf, L (f)iL2 v(µ−1/2) 6 −λLkf − πL(f )k 2 L2 v(hviγ/2µ−1/2) ; (ii) Let E = L2 x,v µ−1/2 or E = L∞ x,v hviβµ−1/2
with β > 3/2. The linear perturbed operator G= L− v · ∇x generates a strongly continuous semigroup
SG(t) on E and there exist CE, λE > 0 such that
where πL is the orthogonal projection onto Ker(L) in L2v µ−1/2
and ΠG is the
orthogonal projection onto Ker(G) in L2 x,v µ
−1/2.
The constants λL, CE and λE are explicit and depend on N , E, the different masses
mi and the collision kernels.
We now state the results we obtain for the full nonlinear equation.
Theorem 2.2. Let the collision kernels Bij satisfy assumptions (H1)− (H4) and
let E = L1
vL∞x hvik with k > k0, where k0 is the minimal integer such that
(2.1) Ck = 2 k + 2 1− max i,j |mi−mj| mi+mj k+22 + 1− max i,j |mi−mj| mi+mj k+22 1− max i,j |mi−mj| mi+mj max i,j 4πb∞ ij lbij < 1. where lbij and b∞ij are angular kernel constants (1.9).
Then there exist ηE, CE and λE > 0 such that for any F0 = µ + f0 > 0 satisfying
the conservation of mass, momentum and energy (1.5) with u∞ = 0 and θ∞= 1, if
kF0− µk 6 ηE
then there exists a unique solution F = µ + f in E to the multi-species Boltzmann equation (1.1) with initial data f0. Moreover, F is non-negative, satisfies the
con-servation laws and
∀t > 0, kF − µkE 6 CEe−λEtkF0− µkE.
The constants are explicit and only depend on N , k, the different masses mi and the
collision kernels.
Remark 2.3. We make a few comments about the theorem above.
(1) As mentioned in the introduction, µ can be replaced by any global equilibrium M(ci,∞, u∞, θ∞). Moreover, as we shall see in Section 6, the natural weight
for this theory is the one associated to the conservation of individual masses and total energy: (1 + mk/2i |v|k)16i6N. This weight is equivalent to hvik
and we keep the latter weight to work without vector-valued masses outside Subsection 6.1.2.
(2) The uniqueness has to be understood in a perturbative regime, that is among the solutions that can be written under the form F = µ + f . We do not give a global uniqueness in L1
vL∞x hvik (as proved in [22] for the single-species
Boltzmann equation).
(3) As a by-product of the proof of uniqueness, we prove that the spectral-gap estimate of Theorem 2.1 also holds for E = L1
vL∞x hvik with k > k0.
(4) In the case of identical masses and hard sphere collision kernels (b = 1) we recover Ck = 4/(k + 2) and thus k0 = 2 which has recently been obtained in
3. Spectral gap for the linear operator in L2
v µ−1/2
3.1. First properties of the linear multi-species Boltzmann operator. We start by describing some properties of the linear multi-species Boltzmann operator L= (Li)16i6N. First recall
Li(f ) = N X j=1 Lij(fi, fj), 1≤ i ≤ N, with Lij(fi, fj) = Qij(µi, fj) + Qij(fi, µj) = Z R3×S2 Bij(|v − v∗|, cos(θ)) µ0∗jfi0+ µ0ifj0∗− µ∗jfi− µifj∗ dv∗dσ,
where we have used µ0∗ i µ
0 j = µ
∗
iµj for any i, j, which follows from the laws of elastic
collisions (1.2).
Some results about the kernel of L have recently been obtained [13] in the case of multi-species having same mass (mi = mj). Their proofs are directly applicable in
the case of different masses, and we therefore refer to their work for detailed proofs. L is a self-adjoint operator in L2 v µ −1/2 with hf, L(f)iL2 v(µ−1/2) = 0 if and only if f belongs to Ker(L). Ker (L) = Spanφ1(v), . . . , φN +4(v) ,
where (φi)16i6N +4 is an orthonormal basis of Ker (L) in L 2 v µ
−1/2. More precisely,
if we denote πL the orthogonal projection onto Ker (L) in L2v µ −1/2: πL(f ) = N +4 X k=1 Z R3 hf(v), φk(v)iµ−1/2 dv φk(v), and ek= (δik)16i6N , we can write (3.1) φk(v) = √1 c∞,k µkek, 1 6 k 6 N φk(v) = vk−N N P i=1 mic∞,i 1/2 (miµi)16i6N, N + 1 6 k 6 N + 3. φN +4(v) = 1 N P i=1 c∞,i 1/2 |v|2− 3m−1 i √ 6 miµi ! 16i6N . Finally, we denote π⊥
L = Id− πL. The projection πL(f (t, x,·))(v) of f(t, x, v) onto
the kernel of L is called its fluid part whereas π⊥
L(f ) is its microscopic part.
L can be written under the following form
where ν = (νi)16i6N is a multiplicative operator called the collision frequency (3.3) νi(v) = N X j=1 νij(v), with νij(v) = CijΦ Z R3×S2 bij(cos θ)|v − v∗|γµj(v∗) dσdv∗.
Each of the νij could be seen as the collision frequency ν(v) of a single-species
Boltzmann kernel with kernel Bij. It is well-known (for instance [11][12][37][22])
that under our assumptions: ν(v) ∼ (1 + |v|γ)∼ hviγ. This means that for all i, j
there exist νij(0), νij(1) > 0 (they are explicit, see the references above) such that ∀v ∈ R3, ν(0) ij (1 +|v| γ ) 6 νij(v) 6 ν (1) ij (1 +|v| γ) ,
Every constant being strictly positive, the following lemma follows straightforwardly. Lemma 3.1. There exists a constant β > 0, and for all i in {1, . . . , N} there exist νi(0), νi(1) > 0 such that (3.4) ∀v ∈ R3 , νi(0)(1 +|v| γ ) 6 νi(v) 6 νi(1)(1 +|v| γ ) . Thus, we get the following relation between the collision frequencies
(3.5) ∀v ∈ R3
, νi(v) 6 βνii(v).
Remark 3.2. Estimate (3.5) is a crucial step in the proof of Lemma 3.4. In [13] the additional assumption Bij 6 CBii for a constant C > 0 has been used in order
to get (3.5). We want to point out that despite of even having different masses to handle, we manage to get rid of this assumption. The prize we have to pay is a slightly more restrictive assumption on the collision kernel B in assumption (H3).
Next we decompose the operator L into its mono-species part Lm = (Lm i )16i6N
and its bi-species part Lb = (Lb
i)16i6N according to L= Lm+ Lb, Lm i (fi) = Lii(fi, fi), Lbi(f ) = X j6=i Lij(fi, fj). (3.6)
Thus f can be written as
(3.7) f = πLm(f ) + πL⊥m(f ),
where πLm is the orthogonal projection on Ker(Lm) with respect to L2v µ−1/2, and
πL⊥m := (1− πLm) .
By employing the standard change of variables, the Dirichlet forms of Lm and Lb
have the form hf, Lm(f ) iL2 v(µ−1/2) = − 1 4 N X i=1 Z R6×S2 Biiµiµ∗i Aiifiµ−1i , fiµ−1i 2 , (3.8) f, Lb(f ) L2 v(µ−1/2) = − 1 4 N X i=1 X j6=i Z R6×S2 Bijµiµ∗j Aijfiµ−1i , fjµ−1j 2 , (3.9)
with the shorthands Aijfiµ−1i , fjµ−1j := fiµ−1i 0 + fjµ−1j 0∗ − fiµ−1i − fjµ−1j ∗ . (3.10)
Since Lm describes a multi-species operator when all the cross-interactions are
null,
πLm(f )i = miµi(v)(ai(t, x) + ui(t, x)· v + ei(t, x)|v|2), 1 6 i 6 N,
(3.11)
where ai ∈ R, ui ∈ R3 and ei ∈ R are the coordinates of πLm(f ) with respect to a
5N -dimensional basis, while
πL(f )i = miµi(v)(ai(t, x) + u(t, x)· v + e(t, x)|v|2) 1 6 i 6 N,
(3.12)
where ai ∈ R, u ∈ R3 and e ∈ R are the coordinates of πL(f ) with respect to an
(N + 4)-dimensional basis. Finally, since Z R3 µidv = ci, Z R3 µi|v|2 dv = 3cim−1i , Z R3 µi|v|4dv = 15cim−2i , (3.13)
the following moment identities hold for ai, ui, ei defined in (3.11)
Z R3 fidv = ci(miai+ 3ei), Z R3 fiv dv = ciui, (3.14) Z R3 fi|v|2dv = ci(3ai+ 15eim−1i ).
3.2. Explicit spectral gap. This subsection is devoted to the proof of the follow-ing constructive spectral gap estimate for the multi-species linear operator L with different masses.
Theorem 3.3. Let the collision kernels Bij satisfy assumptions (H1)-(H4). Then
there exists an explicit constant λL> 0 such that
hf, L(f)iL2
v(µ−1/2) 6 −λLkf − πL(f )k 2 L2
v(hviγ/2µ−1/2) ∀f ∈ Dom(L),
where λL depends on the properties of the collision kernel, the number of species N
and the different masses.
The next two lemmas are crucial for the proof of Theorem 3.3, generalizing the strategy of [13] to the case of different masses. The key idea is to decompose L into L = Lm + Lb (see (3.6)), and to derive separately a spectral-gap estimate for the
type estimate for the bi-species part Lb on Ker(Lm) (see Lemma 3.5) measured in
terms of the following functional E : Ker(Lm) → R+, E(f) := N X i,j=1 u (f ) i − u (f ) j 2 +e(f )i − e(f )j 2 , where for a fixed f ∈ Ker(Lm), u(f )
i and e (f )
i describe the coordinates of the ith
component of f with respect to the basis defined in (3.11). To lighten computations, we introduce the following Hilbert space H := L2
v ν1/2µ−1/2, which is equivalent to L2 v hviγ/2µ −1/2: H = f ∈ L2 v(µ −1/2 ) :kfk2 H = N X i=1 Z R3 f2 iνiµ−1i dv <∞ . (3.15)
Lemma 3.4. For all f in Dom(Lm) there exists an explicit constant C
1 > 0, such that hf, Lm (f )iL2 v(µ−1/2) 6 −C1kf − πL m(f )k2 L2 v(hviγ/2µ−1/2),
where C1 depends on the properties of the collision kernel, the number of species N
and the different masses.
Proof. By [30, Theorem 1.1 and Remark 1 below it] together with the shorthand introduced in (3.10), 1 4 Z R6×S2 Bii Aiifiµ−1i , fiµ−1i 2 µiµ∗i dvdv∗dσ≥ λmc∞,i Z R3 (fi− πLm(f )i)2νiiµ−1i dv,
where λm > 0 depends on the properties of the collision kernel, the number of species
N and the different masses. Summing this estimate over i = 1, . . . , N and employing (3.9) yields (3.16) − hf, Lm(f ) iL2 v(µ−1/2) ≥ λ m N X i=1 c∞,i Z R3 (fi− πLm(fi))2 νii µi dv.
Now we can estimate νii in terms of νi by using (3.5), and plugging this bound into
(3.16) together with the fact that H is equivalent to L2
v hviγ/2µ−1/2 finishes the
proof.
Lemma 3.5. For all f in Ker(Lm)∩ Dom(Lb) there exists an explicit C
2 > 0 such
that
f, Lb(f )
L2
v(µ−1/2) 6 −C2E(f),
with the functional E defined by E : Ker(Lm) → R+, E(f) := N X i,j=1 u (f ) i − u (f ) j 2 +e(f )i − e(f )j 2 , (3.17)
where for fixed f ∈ Ker(Lm) it holds that u(f ) i , e
(f )
i describe the coordinates of the ith
component of f with respect to the basis defined in (3.11), and C2 > 0 is defined in
Remark 3.6. Note that for f in Ker(Lm) it holds that
E(f) = 0 ⇔ f ∈ Ker(Lb),
since Ker(L) = Ker(Lm)∩ Ker(Lb). This fact together with a multi-species version
of the H-theorem show that the left-hand side of the estimate in Lemma 3.5 is null if and only if the right-hand side is null.
Proof. Let f ∈ Ker(Lm)∩ Dom(Lb). Writing f in the form (3.11) and applying the
microscopic conservation laws (1.2) yields
Aij[fiµ−1i , fjµ−1j ] = mi(ui− uj)· (v0− v) + mi(ei− ej)(|v0|2− |v|2), and thus −hf,Lb(f )i L2 v(µ−1/2) = 1 4 N X i,j=1 j6=i m2i Z R6×S2 Bij(ui− uj)· (v0− v) + (ei− ej)(|v0|2− |v|2) 2 µiµ∗j.
Using the symmetry of Bij and of µiµ∗j together with the oddity of the function
G(v, v∗, σ) = Bij(ui− uj)· (v0− v)(|v0|2− |v|2) with respect to (v, v∗, σ) yields that
the mixed term in the square of the integral above vanishes. Thus we obtain −hf,Lb (f )iL2 v(µ−1/2) = 1 4 N X i,j=1 j6=i m2i Z R6×S2 Bij (3.18) × |(ui − uj)· (v0− v)|2+ (ei− ej)2(|v0|2− |v|2)2µiµ∗j dvdv∗dσ.
We claim that the following holds Z R6×S2 Bij((ui−uj)·(v0−v))2µiµ∗jdvdv∗dσ = |ui− uj|2 3 Z R6×S2 Bij|v−v0|2µiµ∗jdvdv∗dσ.
To prove this identity, we write ui,k and vk for the kth component of the vectors ui
and v, respectively. The change of variables (vk, vk∗, σk) 7→ −(vk, vk∗, σk) for fixed k
leaves Bij, µi, and µ∗j unchanged but v 0 k7→ −v 0 k, such that Z R6×S2 Bijvk0v`µiµ∗j dvdv∗dσ = 0 for `6= k. Moreover, Z R6×S2 Bijvkv`µiµ∗j dvdv∗dσ = 0 for `6= k,
since the integrand is odd. Thus, Z R6×S2 Bij((ui− uj)· (v0− v))2µiµ∗j dvdv∗dσ = 3 X k,`=1
(ui,k − uj,k)(ui,`− uj,`)
Z R6×S2 Bij(v0k− vk)(v0`− v`)µiµ∗j dvdv∗dσ = 3 X k=1 (ui,k − uj,k)2 Z R6×S2 Bij(vk− vk0) 2µ iµ∗j dvdv∗dσ.
Since the integral is independent of k, we get Z R6×S2 Bij((ui− uj)· (v0− v))2µiµ∗j dvdv∗dσ = 1 3 3 X k=1 (ui,k − uj,k)2 Z R6×S2 Bij|v − v0|2µiµ∗j dvdv∗dσ,
which proves the claim.
This implies that for all f in Ker(Lm)∩ Dom(Lb) it holds that
hf, Lb(f )
iL2
v(µ−1/2) 6 −C2E(f),
where E(·) is defined in (3.17) and (3.19) C2 = 1 41≤i,j≤nmin Z R6×S2 m2iBijmin 1 3|v − v 0 |2 , (|v0 |2 − |v|2 )2 µiµ∗j dvdv∗dσ.
The last part is to prove that C2 > 0. For this we note that the integrand of (3.19)
vanishes if and only if |v0
| = |v|. However, the set
X ={(v, v∗, σ)∈ R3× R3× S2 :|v0| = |v|}
is closed since it is the pre-image of {0} of the function H(v, v∗, σ) = |v0|2 − |v|2
which is continuous. Now Xc is open and nonempty and thus has positive Lebesgue
measure, and since the integrand in (3.19) is positive on Xc, we get that C 2 > 0,
which finishes the proof.
Proof of Theorem 3.3. The proof will be performed in 4 steps. To lighten notation, we will use the following shorthands for f ∈ Dom(L):
fk = πLm(f ), f⊥= f − fk, hki = µ−1i fik, h⊥i = µ−1i h⊥i .
(3.20)
Step 1 : Absorption of the orthogonal part. The nonnegativity of −hf, Lb(f )i
L2
v(µ−1/2) ≥ 0 and Lemma 3.4 imply that
−h(f, L(f)iL2 v(µ−1/2) ≥ C1kf − f k k2 H− ηhf, Lb(f )iL2 v(µ−1/2), (3.21)
where η ∈ (0, 1] and C1 > 0 was defined in Lemma3.4. Now it holds that
Aij[hi, hj]2 ≥ 1 2Aij[h k i, h k j] 2 − Aij[h⊥i , h ⊥ j ] 2,
and plugging this into (3.9) and (3.21) implies
−hf, L(f))iL2 v(µ−1/2) ≥ C1kf ⊥ k2 H+ η 8 N X i=1 X j6=i Z R6×S2 BijAij[h k i, h k j] 2 µiµ∗j dvdv∗dσ −η4 N X i=1 N X j6=i Z R6×S2 BijAij[h⊥i , h ⊥ j ] 2µ iµ∗j dvdv∗dσ. (3.22)
Now we prove that (up to a small factor) the last term on the right-hand side can be estimated from below bykf⊥
k2
variables (v, v∗) → (v∗, v) together with i ↔ j and (v, v∗) → (v0, v∗0), and by using the identity µiµ∗j = µ 0 iµ 0∗ j we obtain N X i=1 X j6=i Z R6×S2 BijAij[h⊥i , h ⊥ j] 2µ iµ∗j dvdv∗dσ ≤ 4 N X i=1 X j6=i Z R6×S2 Bij ((h⊥i ) 0 )2+ ((h⊥ j ) 0∗ )2+ (h⊥ i ) 2+ ((h⊥ j) ∗ )2µ iµ∗j dvdv∗dσ ≤ 16 N X i=1 X j6=i Z R6×S2 Bij(h⊥i ) 2 µiµ∗j dvdv∗dσ.
Taking into account the definition (3.3) of νi, we get for the last term on the
right-hand side of (3.22) −η 4 N X i=1 X j6=i Z R6×S2 BijAij[h⊥i , h ⊥ j ] 2µ iµ∗j dvdv∗dσ ≥ −4η N X i,j=1 Z R6×S2 Bij(fi⊥)2µ ∗ jµ −1 i dvdv∗dσ ≥ −4η N X i=1 Z R3 (fi⊥)2νiµ−1i dv =−4ηkf ⊥ k2 H. Finally (3.22) yields −hf, L(f)iL2 v(µ−1/2) ≥ (C1− 4η) f − fk 2 H +η 8 N X i=1 X j6=i Z R6×S2 BijAij h hki, hkji2µiµ∗j dvdv∗dσ. Thus hf, L(f)iL2 v(µ−1/2) 6 −(C1− 4η) f − fk 2 H+ η 2hf k , Lb(fk)iL2 v(µ−1/2), (3.23) where 0 < η 6 min{1, C1/8}.
Step 2 : Estimate for for the remaining part. Due to Lemma3.5 there exists an explicit C2 > 0 such that
fk
, Lb(fk)
L2
v(µ−1/2) 6 −C2E f k .
Step 3 : Estimate for the momentum and energy differences. We need to find a relation between E(fk),
f − fk and kf − πL(f )k respectively. To this end, we decompose f = fk+ f⊥ recalling that fk = π
Lm(f ) and f⊥ = f − fk.
Using an arbitrary orthonormal basis (ψk)16k65N of Ker(Lm) in L2 v µ −1/2, we first show that (3.24) kf − πL(f )k2H≤ 2kf ⊥ k2 H+ k0 kfk k2 L2 v(µ−1/2) − kπL(f )k 2 L2 v(µ−1/2) , where k0 = 10N max1≤k,`≤5N|hψk, ψ`iH|.
To this end, we start with
(3.25) kf − πL(f )k2H ≤ 2 kf⊥kH2 +kfk− πL(f )k2H.
Denoting the last term by g := fk−π
L(f )∈ Ker(Lm) (note that Ker(L)⊂ Ker(Lm))
and using Young’s inequality implies kgk2 H = N X i=1 Z R3 5N X k=1 hg, ψkiL2 v(µ−1/2)ψk,i 2 νi(v) dv = 5N X k,`=1 hg, ψkiL2 v(µ−1/2)hg, ψ`iLv2(µ−1/2)hψk, ψ`iH ≤ 1 21≤k,`≤5Nmax |hψk, ψ`iH| 5N X k,`=1 hg, ψki2L2 v(µ−1/2) + hg, ψ`i 2 L2 v(µ−1/2) = 5N max 1≤k,`≤5N|hψk, ψ`iH| kgk 2 L2 v(µ−1/2). Thus, (3.25) implies kf − πL(f )k2H ≤ 2kf ⊥ k2 H+ 10N max 1≤k,`≤5N|hψk, ψ`iH| kf k − πL(f )k2L2 v(µ−1/2).
Now Ker(L)⊂ Ker(Lm) implies π
LmπL= πL, thus kfk − πL(f )k2L2 v(µ−1/2) = kf k k2 L2 v(µ−1/2) − kπL(f )k 2 L2 v(µ−1/2),
which indeed yields (3.24).
Now the moment identities (3.13) and (3.14) yield
kfkk2 L2 v(µ−1/2) = N X i=1 c∞,i(m2ia2i + mi|ui|2+ 15e2i + 6miaiei), and kπL(f )k2L2 v(µ−1/2) = N +4 X j=1 hf, φji2L2 v(µ−1/2) = N X i=1 c∞,i(miai+ 3ei)2+ ρ∞ N X i=1 ρ∞,i ρ∞ ui 2 + 6c∞ N X i=1 c∞,i c∞ ei !2 , where φj
16j6N +4 is the orthonormal basis of Ker(L) in L 2 v(µ
−1/2) introduced in
(3.1).
Inserting these expressions into (3.24), we conclude that
kf − πL(f )k2H ≤ 2kf − f k k2 H+ k0ρ∞ N X i=1 ρ∞,i ρ∞ |u i|2− N X i=1 ρ∞,i ρ∞ ui 2 + 6k0c∞ N X i=1 c∞,i c∞ e2 i − N X i=1 c∞,i c∞ ei !2 .
The next step is to prove that the following estimates hold: I1 := N X i=1 ρ∞,i ρ∞ |u i|2− N X i=1 ρ∞,i ρ∞ ui 2 ≤ N X i,j=1 |ui− uj|2, (3.26) I2 := N X i=1 c∞,i c∞ e2 i − N X i=1 c∞,i c∞ ei !2 ≤ N X i,j=1 (ei− ej)2. (3.27)
Note that we only need to prove the estimate for I1, since the arguments for I2 are
exactly the same. In order to handle the expression I1, we define for u = (ui)16i6N
and v = (vi)16i6N ∈ R3N the following scalar product on R3N with corresponding
norm hu, viρ= N X i=1 ρ∞,i ρ∞ ui· vi, kukρ=hu, ui1/2ρ ,
where ui · vi denotes the standard Euclidean scalar product in R3. Note that the
vector 1 = (1, . . . , 1)∈ R3N satisfiesk1k
ρ= 1. Now we use the following elementary
identity
kuk2
ρ− hu, 1i2ρ=ku − hu, 1iρ1k2ρ,
which can be written as I1 = N X i=1 ρ∞,i ρ∞ |u i|2− N X i=1 ρ∞,i ρ∞ ui 2 = N X i=1 ρ∞,i ρ∞ ui− N X j=1 ρ∞,j ρ∞ uj 2 . By using the fact that PN
j=1ρ∞,j = ρ∞, we get I1 = N X i=1 ρ∞,i ρ∞ 1− ρ∞,i ρ∞ ui− X j6=i ρ∞,j ρ∞ uj 2 = N X i=1 ρ∞,i ρ∞ X j6=i ρ∞,j ρ∞ (ui− uj) 2 . Inserting the additional factor (P
j6=i
ρ∞,k/ρ∞)2 leads to a convex combination of λj
such that P j6=iλj = 1: I1 = N X i=1 ρ∞,i ρ∞ X k6=i ρ∞,k ρ∞ !2 P j6=i(ρ∞,j/ρ∞)(ui− uj) P k6=iρ∞,k/ρ∞ 2 = N X i=1 ρ∞,i ρ∞ X k6=i ρ∞,k ρ∞ !2 X j6=i λj(ui− uj) 2 ,
where λj = (ρ∞,j/ρ∞)(Pk6=i(ρ∞,k/ρ∞))−1. Thus we can apply Jensen’s inequality
to this convex combination and obtain I1 = N X i=1 ρ∞,i ρ∞ X k6=i ρ∞,k ρ∞ !2 X j6=i λj(ui− uj) 2 ≤ N X i=1 ρ∞,i ρ∞ X k6=i ρ∞,k ρ∞ !2 X j6=i λj|ui− uj|2.
Finally, we can estimate the right-hand side easily by using the definition of the λj
and that ρ∞,j ≤ ρ∞ to obtain
I1 6 N X i=1 ρ∞,i ρ∞ 1− ρ∞,i ρ∞ X j6=i ρ∞,j ρ∞ |ui− uj| 2 ≤ N X i,j=1 |ui− uj|2.
For I2 in (3.27) exactly the same calculations hold. This implies that −E(fk ) 6−C3 kf − πL(f )k2H− 2kf − f k k2 H, (3.28) where C3 = 1/Ck > 0, with Ck = 10N max 1≤k,`≤5N N X i=1 Z R3 ψk,iψ`,iνidv max{ρ∞, 6c∞} ,
recalling that (ψk)16k65N is an arbitrary orthonormal basis of Ker(L
m) in L2
v µ−1/2.
Step 4: End of the proof.
Putting together (3.23), Lemma 3.5, and (3.28) yields hf, L(f)iL2 v(µ−1/2) 6 −(C1− 4η)kf − f k k2 H− C2/2E(fk) 6− (C1− 4η − C2C3η)kf − fkkH2 − (C2C3η)/2kf − πL(f )k2H.
The first term on the right-hand side is nonnegative if we choose 0 < η 6 min{1, C1/(4 + C2C3)} ,
and the desired spectral-gap estimate follows with λL = (C2C3C4η)/2, where the
ad-ditional constant C4 > 0 takes care of the fact thatH is equivalent to L2v hviγ/2µ −1/2.
Remark 3.7. (1) We obtain the following relation between the spectral-gap con-stant λ derived for same masses mi = mj for 1 6 i, j 6 N in [13, Theorem
3] and our new constant λL for different masses in Theorem 3.3 :
λL= λ min 16i6Nm 2 i 6ρ∞ max{ρ∞, 6c∞} ,
where ρ∞ = PNi=1mic∞,i and c∞ = PNi=1c∞,i. Thus, increasing the
differ-ence between the masses mi makes the the spectral-gap constant λL smaller,
while in the special case of identical masses the two spectral-gap constants λ and λL are equal.
(2) Furthermore, the spectral-gap result of Theorem 3.3 only holds for a finite number of species 1 6 N < ∞, since for N → ∞ we get that λL → ∞. It
remains an open problem whether or not it is possible to extend the result of Theorem 3.3 to the limit N → ∞.
4. L2
theory for the linear part with maxwellian weight
This section is devoted to the study of the linear perturbed operator G = L−v·∇x
in L2 x,v µ
−1/2, which is the natural space for L. We shall show that G generates a
Theorem 4.1. We assume that assumptions (H1) − (H4) hold for the collision kernel. Then the linear perturbed operator G = L− v · ∇x generates a strongly
continuous semigroup SG(t) on L2x,v µ
−1/2 which satisfies
∀t > 0, kSG(t) (Id− ΠG)kL2
x,v(µ−1/2) 6 CGe −λGt,
where ΠG is the orthogonal projection onto Ker(G) in L2x,v µ −1/2.
The constants CG, λG > 0 are explicit and depend on N , the different masses mi
and the collision kernels.
Let us first make an important remark about ΠG. Note that G(f ) = 0 means
∀i ∈ {1, . . . , N} , ∀(x, v) ∈ T3× R3, v· ∇
xfi(x, v) = Li(f (x,·))(v)
Multiplying by µ−1i (v)fi(x, v) and integrating over T3× R3 implies
0 = Z T3 hLi(f (x,·)), fi(x,·)iL2 v µ−1/2i dx
and therefore by summing over i in {1, . . . , N} 0 =
Z
T3
hL(f(x, ·)), f(x, ·)iL2
v(µ−1/2) dx.
The integrand is nonpositive thanks to the spectral gap of L and hence ∀x ∈ T3,
∀v ∈ R3, f(x, v) = π
L(f (x,·))(v)
and therefore L(f (x,·)) = 0. The latter further implies that v · ∇xf(x, v) = 0 which
in turn implies that f does not depend on x [9, Lemma B.2]. We can thus define the projection in L2
x,v µ
−1/2 onto the kernel of G
(4.1) ΠG(f ) = N +4 X k=1 Z T3×R3 hf(x, v), φk(v)iµ−1/2 dxdv φk(v),
where the φk were defined in (3.1). Again we define Π⊥G = Id − ΠG. Note that
Π⊥
G(f ) = 0 amounts to saying that f satifies the multi-species perturbed conservation
laws (1.8), i.e. null individual mass, sum of momentum and sum of energy.
In Subsection 4.1, we show the key lemma of the proof that is the a priori control of the fluid part of SG(t) by its orthogonal part, thus recovering some coercivity
for G in the set of solutions to the linear perturbed equation. Subsection 4.2 is dedicated to the proof of Theorem 4.1.
4.1. A priori control of the fluid part by the microscopic part. As seen in the previous section, the operator L is only coercive on the orthogonal part. The key argument is to show that we recover some coercivity for solutions to the differential equation. Namely, that for these specific functions, the microscopic part controls the fluid part. This is the purpose of the next lemma
Lemma 4.2. Let f0(x, v) and g(t, x, v) be in L2x,v µ−1/2
such that ΠG(f0) =
ΠG(g) = 0. Suppose that f (t, x, v) in L2x,v µ
−1/2 is solution to the equation
(4.2) ∂tf = L (f )− v · ∇xf + g
with initial value f0 and satisfying the multi-species conservation laws. Then there
exist an explicit C⊥> 0 and a function Nf(t) such that for all t > 0
(i) |Nf(t)| 6 C⊥kf(t)k2L2 x,v(µ−1/2); (ii) Z t 0 kπL(f )k2L2 x,v(µ−1/2) ds 6Nf(t)− Nf(0) + C⊥ Z t 0 π⊥ L(f ) 2 L2 x,v(µ−1/2) ds + C⊥ Z t 0 kgkL2 x,v(µ−1/2) ds.
The constant C⊥ is independent of f and g.
The methods of the proof are a technical adaptation of the method proposed in [15] in the case of bounded domain with diffusive boundary conditions. The description of Ker(L) associated with the global equilibrium µ is given by orthogonal functions in L2
v but that are not of norm one. Unlike [15] where only mass conservation holds
but boundary conditions overcome the lack of conservation laws, we strongly need the conservation of mass, momentum and energy.
Proof of Lemma 4.2. We recall (3.1) the definition of πL(f ) = (πi(f ))16i6N and we
define (ai(t, x))16i6N, b(t, x) and c(t, x) to be the coordinates of πL(f ):
(4.3) ∀ 1 6 i 6 N, πi(f )(t, x, v) = ai(t, x) + b(t, x)· v + c(t, x)|v 2| − 3m−1 i 2 miµi(v).
Note that we are working with an orthogonal but not orthonormal basis of Ker(L) in L2
x,v(µ
−1/2) in order to lighten computations. We will denote by ρ
i the mass of
miµi.
The key idea of the proof is to choose suitable test functions ψ = (ψi)16i6N in
H1
x,v that will catch the elliptic regularity of ai, b and c and estimate them.
For a test function ψ = ψ(t, x, v) integrated against the differential equation (4.2) we have by Green’s formula on each coordinate
Z t 0 d dt Z T3×R3 hψ, fi1dxdvds = Z T3×R3 hψ(t), f(t)i1dxdv− Z T3×R3 hψ0, f0i1dxdv = Z t 0 Z T3×R3 hf, ∂tψi1 dxdvds + Z t 0 Z T3×R3 hL (f) , ψi1dxdvds + N X i=1 Z t 0 Z T3×R3 fiv· ∇xψidxdvds + Z t 0 Z T3×R3 hψ, gi1dxdvds.
We decompose f = πL(f ) + πL⊥(f ) in the term involving v· ∇x and use the fact that
L(f ) = L[π⊥
L(f )] to obtain the weak formulation
(4.4) − N X i=1 Z t 0 Z T3×R3 πi(f )v· ∇xψidxdvds = Ψ1(t) + Ψ2(t) + Ψ3(t) + Ψ4(t) + Ψ5(t)
with the following definitions Ψ1(t) = Z T3×R3 hψ0, f0i1dxdv− Z T3×R3 hψ(t), f(t)i1dxdv, (4.5) Ψ2(t) = N X i=1 Z t 0 Z T3×R3 πL⊥(f )iv· ∇xψidxdvds, (4.6) Ψ3(t) = N X i=1 Z t 0 Z T3×R3 L πL⊥(f ) iψidxdvds, (4.7) Ψ4(t) = N X i=1 Z t 0 Z T3×R3 fi∂sψidxdvds, (4.8) Ψ5(t) = Z t 0 Z T3×R3 hψ, gi1 dxdvds. (4.9)
For each of the functions a = (ai)16i6N, b and c, we construct a ψ such that the
left-hand side of (4.4) is exactly the L2
x-norm of the function and the rest of the
proof is estimating the four different terms Ψi(t). Note that Ψ1(t) is already under
the desired form
(4.10) Ψ1(t) = Nf(t)− Nf(0)
with |Nf(s)| 6 C kfk2L2
x,v(µ−1/2) if ψi(x, v)µ 1/2
i (v) is in L2x,v for all i and their norm is
controlled by the one of f (which will be the case in our next choices).
Remark 4.3. The linear perturbed equation (4.2) and the conservation laws are invariant under standard time mollification. We therefore consider for simplicity in the rest of the proof that all functions are smooth in the variable t. Exactly the same estimates can be derived for more general functions and the method would obviously be to study time mollified equation and then take the limit in the smoothing parameter.
For clarity, every positive constant will be denoted by Ck.
Estimate for a = (ai)16i6N. By assumption f preserves the mass which is
equivalent to 0 = Z T3×R3 f(t, x, v) dxdv = Z T3 Z R3 hf(t, x, v), µiµ−1/2 dv dx = Z T3 a(t, x) dx, where we used the fact that µ ∈ Ker(G), f0 ∈ Ker(G)⊥ and the orthogonality of
the basis defined in (4.3). Define a test function ψa = (ψi)16i6N by
ψi(t, x, v) = |v| 2
where
−∆xφi(t, x) = ai(t, x)
and αi > 0 is chosen such that for all 1 6 k 6 3
Z R3 |v|2− αi |v| 2 − 3m−1 i 2 v 2 kµi(v) dv = 0.
The integral over T3 of a
i(t,·) is null and therefore standard elliptic estimate [16]
yields:
(4.11) ∀t > 0, kφi(t)kH2
x 6 C0kai(t)kL2x.
The latter estimate provides both the control of Ψ1 = N (a)
f (t)−N (a)
f (0), as discussed
before, and the control of (4.9), using Cauchy-Schwarz and Young’s inequality,
|Ψ5(t)| 6 C N X i=1 Z t 0 k√ρiφikL2 xkgikL2x,v µ−1/2i ds 6 C1 4 Z t 0 kak2L2 x(ρ1/2) ds + C5 Z t 0 kgk2L2 x,v(µ−1/2) ds, (4.12)
where C1 > 0 is given in (4.13) below and where we defined ρ = (ρi)16i6N the vector
of the masses associated to (miµi)16i6N.
Firstly, we compute the term on the left-hand side of (4.4).
− N X i=1 Z t 0 Z T3×R3 πi(f )v· ∇xψidxdvds =− N X i=1 X 16j,k63 Z t 0 Z T3 ai(s, x) Z R3 |v|2− αi vjvkmiµi(v) dv ∂xj∂xkφidxds − N X i=1 X 16j,k63 Z t 0 Z T3 b(s, x)· Z R3 v |v|2− αi vjvkmiµi(v) dv ∂xj∂xkφidxds − N X i=1 X 16j,k63 Z t 0 Z T3 c(s, x) Z R3 |v|2− αi |v| 2 − 3m−1 i 2 vjvkmiµidv ! ∂xj∂xkφi.
The second term is null as well as the first and last ones when j 6= k thanks to the oddity in v. In the last term when j = k we recover our choice of αi which makes
the last term being null too. It remains the first term when k = j. In this case, the integral in v gives a constant C1 independent of i times ρi. Direct computations
give αi = 10/mi and C1 > 0. It follows
− N X i=1 Z t 0 Z T3×R3 πi(f )v· ∇xψidxdvds = −C1 N X i=1 Z t 0 Z T3 ai(s, x)ρi∆xφi(s, x) dxds = C1 N X i=1 Z t 0 Z T3 a2iρids = C1 Z t 0 ka(s)k2L2 x(ρ1/2) ds. (4.13)
We recall L =−ν(v)+K where K is a bounded operator in L2
v µ−1/2. Moreover,
the H2
x-norm of φi(t, x) is bounded by the L2x-norm of ai(t, x). Multiplying by
µ1/2i (v)µi(v)−1/2 inside the ith integral of Ψ2 (4.6) and of Ψ3 (4.7) a mere
Cauchy-Schwarz inequality yields
∀k ∈ {2, 3} , |Ψk(t)| 6 C N X i=1 Z t 0 k √ρ iaikL2 x πi⊥(f ) L2 x,v µ−1/2i ds 6 C1 4 Z t 0 kak2L2 x(ρ1/2) ds + C2 Z t 0 π⊥L(f ) 2 L2 x,v(µ−1/2) ds. (4.14)
We used Young’s inequality for the last inequality, with C1 defined in (4.13).
It remains to estimate the term with time derivatives (4.8). It reads
Ψ4(t) = N X i=1 Z t 0 Z T3×R3 fi |v|2− αi v · [∂t∇xφi] dxdvds = N X i=1 3 X k=1 Z t 0 Z T3×R3 πi(f ) |v| 2 − αi vk∂t∂xkφidxdvds + N X i=1 Z t 0 Z T3×R3 πi⊥(f ) |v|2− αi v · [∂t∇xφi] dxdvds
Using oddity properties for the first integral on the right-hand side and then Cauchy-Schwarz with the following bound
Z R3 |v|2− αi 2 |v|2µi(v) dv = Cρi < +∞ we get (4.15) |Ψ4(t)| 6 C N X i=1 Z t 0 " 3 X k=1 kρibkkL2 x + π⊥ i (f ) L2 x,v µ−1/2i # k∂t∇xφikL2 x ds. Estimating k∂t∇xφakL2
x will come from elliptic estimates in negative Sobolev
spaces. We use the decomposition of the weak formulation (4.4) between t and t + ε (instead of between 0 and t) with ψ(t, x, v) = φ(x)ei ∈ Hx1, where ei = (δji)16j6N.
We furthermore require that φ(x) has a null integral over T3. ψ only depends on x
and therefore Ψ4(t) = 0. Moreover, multiplying by µi(v)µ−1i (v) in the ith integral of
Ψ3 yields Ψ3(t) = Z t+ε t Z T3 hL(f), µieiiL2 v(µ−1/2)φ(x) dxdvds = 0, by definition of Ker(L).
From the weak formulation (4.4) it therefore remains Z T3×R3 φ(x)hei, f (t + ε)i1dxdv− Z T3×R3 φ(x)hei, f (t)i1dxdv = Z t+ε t Z T3×R3 πi(f )v· ∇xφ(x) dxdvds + Z t+ε t Z T3×R3 πi⊥(f )v· ∇xφ(x) dxdvds + Z t+ε t Z Ω×R3 gi(s, x, v)φ(x) dxdvds which is equal to Z T3 ρi[ai(t + ε)− ai(t)] φ(x) dx = C Z t+ε t Z T3 ρib(s, x)· ∇xφ(x) dxds + Z t+ε t Z T3×R3 πi⊥(f )µi(v)−1/2µi(v)1/2v· ∇xφ(x) + Z t+ε t Z Ω×R3 gi(s, x, v)φ(x) dxdvds,
where C does not depend on i.
Dividing by ρiε and taking the limit as ε goes to 0 yields, after a mere
Cauchy-Schwarz inequality on the right-hand side Z T3 ∂tai(s, x)φ(x) dx 6 C kb(t, x)kL2 x+ πi⊥(f ) L2 x,v µ−1/2i k∇xφ(x)kL2 x +CkgikL2 x,v µ−1/2i kφkL2 x 6 C kb(t, x)kL2 x+ πi⊥(f ) L2 x,v µ−1/2i +kgikL2 x,v µ−1/2i × k∇xφ(x)kL2 x.
We used Poincar´e inequality since φ(x) has a null integral over Td. The latter
inequality is true for all φ in H1
x with a null integral and therefore implies for all
t > 0 (4.16) k∂tai(t, x)k(H1 x) ∗ 6 C kb(t, x)kL2 x + πi⊥(f ) L2 x,v µ−1/2i +kg ikL2 x,v µ−1/2i where (H1 x) ∗
is the dual of the set of functions in H1
x with null integral.
Thanks to the conservation of mass we have that ∂tai(t, x) have a zero integral on
the torus and we can construct Φi(t, x) such that
−∆xΦi(t, x) = ∂tai(t, x)
and by standard elliptic estimate [16]: kΦikH1 x 6k∂taik(H1x) ∗ 6 C kb(t, x)kL2 x + π⊥i (f ) L2 x,v µ−1/2i +kg ikL2 x,v µ−1/2i , where we used (4.16). Combining this estimate with
k∂t∇xφikL2 x = ∇x∆−1∂tai L2 x 6 ∆−1∂tai H1 x =kΦikHx1
we can further control Ψ4 in (4.15) using ρi = √ρi√ρi (4.17) |Ψ4(t)| 6 C5 Z t 0 N X i=1 k√ρibk 2 L2 x+ π⊥i (f ) 2 L2 x,v µ−1/2i +kg ik 2 L2 x,v µ−1/2i ! ds.
We now plug (4.13), (4.10), (4.14), (4.17) and (4.12) into (4.4) Z t 0 kak2L2 x(ρ1/2) ds 6N (a) f (t)− N (a) f (0) + Ca,b Z t 0 kbk2L2 x(ρ1/2) ds + Ca Z t 0 h πL⊥(f ) 2 L2 x,v(µ−1/2) + kgk 2 L2 x,v(µ−1/2) i ds. (4.18)
Estimate forb. The choice of function to integrate against to deal with the b term is more involved technically. We emphasize that b(t, x) is a vector (b1(t, x), b2(t, x), b3(t, x)),
thus we used the obvious short-hand notation kbk2L2 x(ρ1/2) = N X i=1 3 X k=1 k√ρibkk 2 L2 x.
Fix J in {1, 2, 3} and the conservation of momentum implies that for all t > 0 Z
T3
bJ(t, x) dx = 0.
Define ψbJ(t, x, v) = (ψiJ(t, x, v))16i6N with
ψiJ(t, x, v) = 3 X j=1 ϕ(J )ij (t, x, v), with ϕ(J )ij (t, x, v) = |v|2vjvJ∂xjφJ(t, x)− 7 2mi vj2− m −1 i ∂xJφJ(t, x), if j 6= J 7 2mi v2J− m −1 i ∂xJφJ(t, x), if j = J. where −∆xφJ(t, x) = bJ(t, x).
Since it will be important, we emphasize here that for all j 6= k (4.19) Z R3 v2 j − m −1 i µi(v) dv = 0 and Z R3 v2 j − m −1 i v 2 kµi(v) dv = 0.
The null integral of bJ implies by standard elliptic estimate [16]
(4.20) ∀t > 0, kφJ(t)kH2
x 6 C0kbJ(t)kL2x.
Again, this estimate provides the control of Ψ1(t) = N (J ) f (t)− N (J ) f (0) and of Ψ5(t) as in (4.12): (4.21) |Ψ5(t)| 6 C1 4 Z t 0 kbJk2L2 x(ρ1/2) ds + C5 Z t 0 kgk2L2 x,v(µ−1/2) ds,
We start by the left-hand side of (4.4). By oddity, there is neither contribution from any of the ai(s, x) nor from c(s, x). Hence, for all i in {1, . . . , N}
− Z t 0 Z Ω×R3 πi(f )v· ∇xψiJ dxdvds =− X 16k,l63 3 X j=1 j6=J Z t 0 Z Ω bl(s, x) Z R3 v2 vlvkvjvJmiµi(v) dv ∂xk∂xjφJ(s, x) dxds + 7 2mi X 16k,l63 3 X j=1 j6=J Z t 0 Z Ω bl(s, x) Z R3 v2 j − m −1 i vlvkmiµidv ∂xk∂xJφJ dxds − 7 2mi X 16k,l63 Z t 0 Z Ω bl(s, x) Z R3 v2 J − m −1 i vlvkmiµi(v) dv ∂xk∂xJφJ dxds.
The last two integrals on R3 are zero if l 6= k. Moreover, when l = k and l 6= J it is
also zero by (4.19). We compute directly for l = J Z R3 v2 J − m −1 i v 2 Jmiµi(v) dv = 2 m2 i ρi.
The first term is composed by integrals in v of the form Z
R3
|v|2vkvjvlvJµi(v) dv
which is always null unless two indices are equals to the other two. Therefore if j = l then k = J and if j 6= l we only have two options: k = j and l = J or k = l and j = J. Hence, for all i in{1, . . . , N}
− Z t 0 Z Ω×R3 πi(f )v· ∇xψJdxdvds =− 3 X j=1 j6=J Z t 0 Z Ω bJ(s, x)∂xjxjφJ Z R3 |v|2vj2v 2 Jmiµi(v) dv dxds − 3 X j=1 j6=J Z t 0 Z Ω bj(s, x)∂xjxJφJ Z R3 |v|2v2 jv 2 Jmiµi(v) dv dxds + 7 m3 i 3 X j=1 j6=J Z t 0 Z Ω ρibj(s, x)∂xjxJφJ dxds− 7 m3 i Z t 0 Z Ω ρibJ(s, x)∂xJ∂xJφJ(s, x) dxds.
To conlude we compute for j 6= J Z R3 v2 vj2v2Jmiµi(v) dv = 7 m3 i ρi
and it thus only remains the following equality for all i in {1, . . . , N}. − Z t 0 Z Ω×R3 πi(f )v· ∇xψJ dxdvds = − 7 m3 i Z t 0 Z Ω ρibJ(s, x)∆xφJ(s, x) dxds = 7 m3 i Z t 0 k√ρibJk2L2 x ds.
Summing over i yields
(4.22) − N X i=1 Z t 0 Z Ω×R3 πj(f )v· ∇xψJ = 7 m3 i Z t 0 kbJkL2 x(ρ1/2) dxdvds. We recall ρ = (ρi)16i6N.
Then the terms Ψ2 and Ψ3 are dealt with as in (4.14)
(4.23) ∀k ∈ {2, 3} , |Ψk(t)| 6 7 4 Z t 0 kbJk2L2 x(ρ1/2) ds + C2 Z t 0 π⊥L(f ) 2 L2 x,v(µ−1/2) ds.
It remains to estimate Ψ4 which involves time derivative (4.8):
Ψ4(t) = N X i=1 3 X j=1 Z t 0 Z Ω×R3 fi∂tϕ (J ) ij (s, x, v) dxdvds = N X i=1 3 X j=1 Z t 0 Z Ω×R3 π⊥i (f )∂tϕ (J ) ij (s, x, v) dxdvds + N X i=1 3 X j=1 j6=J Z t 0 Z Ω×R3 πi(f )|v| 2 vjvJ∂xjφJ dxdvds + N X i=1 3 X j=1 ± 7 2mi Z t 0 Z Ω×R3 πi(f ) v2j − m −1 j ∂xJφJ dxdvds.
By oddity arguments, only terms in ai(s, x) and c(s, x) can contribute to the last
two terms on the right-hand side. However, j 6= J implies that the second term is zero as well as the contribution of ai(s, x) in the third term thanks to (4.19). Finally,
a Cauchy-Schwarz inequality on both integrals yields as in (4.15)
(4.24) |Ψ4(t)| 6 C N X i=1 Z t 0 kρickL2 x + πi⊥(f ) L2 x,v µ−1/2i k∂t∇xφJkL2 x ds. To estimate k∂t∇xφJkL2
x we follow the idea developed for a(s, x) about negative
Sobolev regularity. We apply the weak formulation (4.4) to a specific function be-tween t and t + ε. The test function is ψ(x, v) = φ(x)vJmwith φ in Hx1 with a zero
integral over T3. Note that ψ does not depend on t so Ψ
4 = 0 and multiplying by
µi(v)µ−1i (v) in the ith integral of Ψ3 yields
Ψ3(t) = Z t 0 Z T3 hL(f), vJ(miµi)16i6NiL2 v(µ−1/2)∂xkφ(x) dxdvds = 0, by definition of Ker(L).
It remains C N X i=1 Z Ω ρi[bJ(t + ε)− bJ(t)] φ(x) dx = N X i=1 Z t+ε t Z Ω×R3 πi(f )vJv· ∇xφ(x) dxdvds + N X i=1 Z t+ε t Z Ω×R3 πi⊥(f )vJv· ∇xφ(x) dxdvds + N X i=1 Z t+ε t Z Ω×R3 givJφ(x) dxdvds.
As for ai(t, x) we divide by ε and take the limit as ε goes to 0. By oddity, the first
integral on the right-hand side only gives terms with ai(s, x) and c(s, x). The other
two integrals are dealt with by a Cauchy-Schwarz inequality and Poincar´e. This yields Z Ω ∂tbJ(t, x)φ(x) dx 6 ChkakL2 x(ρ1/2) + kckL2x(ρ1/2) + π⊥L(f ) L2 x,v(µ−1/2) + k gkL2 x,v(µ−1/2) i k∇xφkL2 x. (4.25)
The latter is true for all φ(x) in H1
x with a null integral over T3. We thus fix t
and apply the inequality above to
−∆xφ(t, x) = ∂tbJ(t, x)
which has a zero integral thanks to the conservation of momentum and obtain k∂t∇xφJk2L2 x = ∇x∆−1∂tbJ 2 L2 x = Z Ω ∇x∆−1∂tbJ ∇xφ(x) dx. We integrate by parts k∂t∇xφJk2L2 x = Z Ω ∂tbJ(t, x)φ(x) dx. At last, we use (4.25) k∂t∇xφJk 2 L2 x 6 ChkakL2 x(ρ1/2) + kckL2x(ρ1/2) + π⊥L(f ) L2 x,v(µ−1/2) + k gkL2 x,v(µ−1/2) i k∇xφkL2 x = ChkakL2 x(ρ1/2) + kckL2x(ρ1/2) + πL⊥(f ) L2 x,v(µ−1/2) + kgkL2x,v(µ−1/2) i ∇x∆−1x ∂tbJ L2 x = ChkakL2 x(ρ1/2) + kckL2x(ρ1/2) + πL⊥(f ) L2 x,v(µ−1/2) + kgkL2x,v(µ−1/2) i k∂t∇xφJkL2 x.
